Monitoring of wind turbines

ABSTRACT

Method and apparatus for determining the deflection or curvature of a rotating blade, such as a wind turbine blade or a helicopter blade. Also, methods and apparatus for establishing an inertial reference system on a rotating blade.

PRIORITY CLAIM

This application is a national stage filing of PCT Patent ApplicationSerial No. PCT/US2009/043856, filed May 13, 2009, and titled MONITORINGOF WIND TURBINES, which claims the benefit of priority to U.S.Provisional Patent Application Ser. No. 61/052,848, filed May 13, 2008and titled MODAL ANALYSIS OF WIND TURBINES, both of which areincorporated herein by reference.

GOVERNMENT RIGHTS

This invention was made with government support under contract numberDE-AC04-94AL85000 awarded by the Department of Energy. The governmenthas certain rights in the invention.

FIELD OF THE INVENTION

The present invention pertains to analysis of structures, and inparticular to static and dynamic analysis of wind turbines.

BACKGROUND OF THE INVENTION

In 2008 the U.S. installed 8,358 MW of new wind capacity making windenergy the fastest growing source of renewable energy. The total U.S.installed wind power increased to 25,170 MW Germany with 23,902 MW ofwind energy was surpassed by the U.S. as the largest producer of windpower in the world. The installation of wind turbines accounted for 42%of all new energy capacity installed in the United States. 5.5 millionhomes can now be served by the wind energy produced in the U.S.

Each of these installed wind turbines represent a significant investmentand a significant resource. As the population of wind turbines increasesand ages, malfunctions can be expected to occur. If such malfunctionscan be predicted prior to happening, then the wind turbine can be shutdown before expensive damage occurs. Further, the local power utilitycompany may be afforded an opportunity to plan for the shutdown of thisresource and the need to replace it while the wind turbine is out ofcommission.

Some goals of a wind turbine rotor health monitoring system can include:(1) estimate mechanical loading, and (2) monitor damage. Estimation ofthe turbine input could lead to more efficient rotor control and aid infuture blade design. Knowledge of the damage state would allow forplanned maintenance and avoid catastrophic failure.

Some of the various embodiments of the present invention described andclaimed hereafter show novel and nonobvious ways of improving a windturbine.

SUMMARY OF THE INVENTIONS

Some aspects of the inventions disclosed herein pertain to methods forestablishing an inertial reference on a rotating system from one or morerelative (i.e., non-inertial) measurements of a rotating member.

Yet other aspects of some embodiments of the present inventions pertainto the use of a function to predict geometry of a rotating member. Asone example, the predicted geometry can be the lateral deflection of themember. The function can include one or more unknown coefficients, eachof which multiplies a different term within the function. In someembodiments, a derivative of the function is calculated, and measureddata is used to calculate the unknown coefficients as they are expressedin the derivative. The same coefficients can be used in the function topredict the geometry of the member.

One aspect of some embodiments of the present invention pertains to amethod of predicting the deflection of a rotating member. The methodfurther includes providing a rotating member attached to a rotatablehub, and a sensor attached to the member along the length. Certainembodiments include an end of the rotating member attached to the hubsuch that the slope of the member at the hub is about zero duringrotation. Some embodiments include expressing the deflection of themember as a function of the length of the member, the function includingat least one unknown coefficient. Yet other embodiments include taking aderivative of the function with respect to length, the derivativeincluding the unknown coefficient or at least two unknown coefficients.In further embodiments, the function is a series expansion having anorder greater than one or an order greater than two. Still otherembodiments include measuring a response of the member with a sensorduring rotation, using the response to calculate the coefficient orcoefficients. Yet further embodiments include converting the measuredresponse by at least one of time-based filtering or numerical scaling.

Another aspect of some embodiments of the present invention pertains toa method of determining the slope of a rotating member. The methodfurther includes providing a flexible member attached to a rotatablehub, with one end being free to deflect. A sensor provides a measurementof velocity or acceleration of the member. Yet other embodiments includeattaching the sensor to the member at a predetermined station along thelength. Still further embodiments include rotating the hub and attachedmember, measuring velocity or acceleration during said rotating, andcalculating a vector of centripetal acceleration from the measurement.Some aspects of the invention include calculating the slope at thepredetermined station from orientation of the vector relative to thedirection of measurement.

Yet another aspect of some embodiments of the present invention includesa method of determining the bending of a rotating member. The methodfurther includes providing a flexible member attached to a rotatable hubwith the other end being free to deflect. A sensor provides a signalaligned with a measurement direction and capable of providing datacorresponding to centripetal acceleration of the sensor. Someembodiments include attaching the sensor to the member along the lengthsuch that the measurement direction stays in a fixed orientationrelative to the bending member. Some aspects of the invention includecalculating a vector of centripetal acceleration from the sensor dataand calculating the slope at the predetermined station from theorientation of the vector relative to the direction of measurement.

Another aspect of the present invention includes a wind turbine having aplurality of blades each attached to a rotating hub. Some embodimentsinclude an acceleration sensor attached to one of said blades andproviding a plurality of signals, each signal corresponding toacceleration in a different direction. Yet other embodiments include acomputer receiving the plurality of signals from the sensor. Stillfurther embodiments include a software algorithm using the signals tocalculate a vector of centripetal acceleration during rotation of thehub and blade. The algorithm determines the angular offset from thecalculated vector to at least one of said signals.

It will be appreciated that the various apparatus and methods describedin this summary section, as well as elsewhere in this application, canbe expressed as a large number of different combinations andsubcombinations. All such useful, novel, and inventive combinations andsubcombinations are contemplated herein, it being recognized that theexplicit expression of each of these myriad combinations is excessiveand unnecessary.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1a is a perspective drawing of a wind turbine according to oneembodiment of the present invention.

FIG. 1b shows graphical representation of a coordinate system used inthis document.

FIG. 1c is a graphical depiction showing the turbine blade loading (L)that causes a deflection (δ) and resulting hub forces (F) and moments(M), on a turbine blade according to one embodiment of the presentinvention.

FIG. 2 is a graphical depiction showing the distributed triaxial anduniaxial DC accelerometer sensor array of the blade of FIG. 1 c.

FIG. 3 is a close-up photographic representation of the 8 m station withtwo triaxial sensors, a uniaxial sensor and a RTD temperature sensor onthe blade of FIG. 1 c.

FIG. 4 is a photographic typical cross section of a sensor array of theblade of FIG. 1 c.

FIG. 5a is a schematic representation of a static blade.

FIG. 5b is a schematic representation of wind loading and the resultantdeflection in the blade of FIG. 5 a.

FIG. 5c includes graphical and mathematical representations of theaccelerations measured in FIG. 5b , and the relationship of the measuredaccelerations to the slope of the blade at a point along the length ofthe blade.

FIG. 6 are a graphical representation of wind loading along the lengthof a blade from the hub to tip and showing deflection, u, of rotatingEuler-Bernoulli beam with a rigid hub caused by the distributed windloading, W, along the span direction, z, of the rotor blade with bendingstiffness El(z) as a function of wind speed, U.

FIG. 7 are graphical representations pertaining to blade loading asfollows: (a) actual flap-wise distributed loading on CX-100 blade acrossthe blade span for wind speeds from 1-30 m/s as predicted by FAST; (b)estimated distributed loading from simplified model derived using SAS©;and (c) difference between actual and estimated distributed loading.

FIG. 8 are graphical representations pertaining to error calculations asfollows: (a) error surface for CX-100 realistic loading case as afunction of wind speed and sensor position; and (b) error functionintegrated over wind range zero to thirty meters per second showing theoverall optimal sensor location at 5.0 meters.

FIG. 9 are graphical representations pertaining to blade deflection andblade loading as follows: (a) difference between true rotor bladedeflection and estimated rotor blade deflection in meters as a functionof rotor span and wind speed; and (b) estimated rotor blade distributeloading as a function of wind speed and rotor span.

DESCRIPTION OF THE PREFERRED EMBODIMENT

For the purposes of promoting an understanding of the principles of theinvention, reference will now be made to the embodiments illustrated inthe drawings and specific language will be used to describe the same. Itwill nevertheless be understood that no limitation of the scope of theinvention is thereby intended, such alterations and furthermodifications in the illustrated device, and such further applicationsof the principles of the invention as illustrated therein beingcontemplated as would normally occur to one skilled in the art to whichthe invention relates. At least one embodiment of the present inventionwill be described and shown, and this application may show and/ordescribe other embodiments of the present invention. It is understoodthat any reference to “the invention” is a reference to an embodiment ofa family of inventions, with no single embodiment including anapparatus, process, or composition that must be included in allembodiments, unless otherwise stated.

The use of an N-series prefix for an element number (NXX.XX) refers toan element that is the same as the non-prefixed element (XX.XX), exceptas shown and described thereafter. As an example, an element 1020.1would be the same as element 20.1, except for those different featuresof element 1020.1 shown and described. Further, common elements andcommon features of related elements are drawn in the same manner indifferent figures, and/or use the same symbology in different figures.As such, it is not necessary to describe the features of 1020.1 and 20.1that are the same, since these common features are apparent to a personof ordinary skill in the related field of technology. Although variousspecific quantities (spatial dimensions, temperatures, pressures, times,force, resistance, current, voltage, concentrations, wavelengths,frequencies, heat transfer coefficients, dimensionless parameters, etc.)may be stated herein, such specific quantities are presented as examplesonly. Further, discussion pertaining to a specific composition ofmatter, that description is by example only, does not limit theapplicability of other species of that composition, nor does it limitthe applicability of other compositions unrelated to the citedcomposition.

Although wind turbines are becoming a mature and serious contributor tothe production of power, there is still much potential to furtherimprove the design, control and performance of turbines. Sandia NationalLaboratories Wind Energy Technology Department (SNL) is currentlydeveloping technologies to produce an advanced smart rotor blade. Onevision of this program is a rotor blade that can (1) detect the loadingproduced by the wind, (2) estimate the transfer of these loads to thegearbox, (3) estimate the backlash loads from the drive-train to therotor blade, (4) adapt the rotor blade aerodynamics with microtabs,ailerons or fast pitch actuators to increase load at low wind speed andshed load at high wind speed, (5) calculate the accumulation of dynamicfatigue cycles, (6) detect damage throughout the blade and (7) verifydesign specifications and predicted performance.

Various embodiments of the invention presented herein pertain totechnology to fulfill some of these requirements, specifically toestimate the loading to the blade, provide an observer for the controlsystem and monitor damage accumulation. Few attempts have been made inprevious work to accomplish these goals with inertial sensors, namelyaccelerometers, as presented in this work. Inertial sensors have thebenefit of directly measuring acceleration, the physical variable usedin the equations of motion of structural dynamic systems. Some of thechallenges of implementing accelerometers include: the signal processingof the higher-order measurement, relationship between the dynamics ofthe structure and the sensor data, constraints imposed by theperformance characteristics of accelerometers, the effects oforientation on the acceleration data, and the estimation of staticloading with a dynamic measurement. Various embodiments of theinventions shown herein offer novel and nonobvious approaches developedto overcome these challenges.

One technology to improve the efficiency of wind turbines is the use ofsmart rotor blades, which can monitor the physical loads being appliedby the wind and then adapt the airfoil for increased energy capture. Forextreme wind and gust events, the airfoil could be changed to reduce theloads to prevent excessive fatigue or catastrophic failure. Knowledge ofthe actual loading to the turbine is also useful for maintenanceplanning and design improvements.

A prototype smart blade is presented that captures flap deflection andtorsional deflection shapes to estimate the loading of the wind over theentire blade. DC type accelerometers are utilized in order to estimatethe loading and deflection from both quasi-steady-state and dynamicevents. Modal properties were measured for the turbine blade infree-free, cantilevered, and assembled boundary conditions to develop amodel for use in estimating the operational forcing functions.

Experimental results showed that blade displacement and rotation couldbe estimated from steady-state loading using DC accelerometers. Dynamicloading to the turbine blade was also estimated through the use of modalfiltering. The results demonstrated that a DC accelerometer sensor arraycould be used to estimate loads and deflections, which are needed toreduce maintenance costs and enable adaptive control of the blade.

Passive and active damage detection methods applied to a previousfatigue to failure test are also covered as well. In-plane measurementsare shown to be sensitive to damage and transverse measurements areuseful load estimation. The active method was capable of compensatingfor temperature changes and then producing an indicative estimate of thedamage.

Wind turbines have only recently been considered as applications forsmart structure and structural health monitoring technologies.Consequently, the literature available on monitoring and control of windturbine rotor blades is somewhat limited. A principle area of researchhas been in the development of actuators for individual blade control.Although no single method has been identified as superior, the objectivehas been to improve the efficiency of the rotor blades by activelycontrolling the aerodynamics of the operational turbine. The currentblades are designed with varying twist from the root to tip so that theblade has an optimized performance over the most productive wind speedsthat are anticipated. The performance of wind turbine rotors isoptimized by balancing the increased power available at low occurrencehigh wind speeds with the decreased power available in high occurrencelow wind speeds.

In addition to design of the rotor blade to achieve specific aerodynamicperformance, most industrial scale wind turbines have the ability toindividually pitch the entire blade thereby additionally changing theperformance of the rotor blade. However, turbulent and gust events,which can produce significant rotor damage or produce significant power,have time constants much faster than the pitch rate of the rotor blade.In addition, the average blade length is around 40 meters, and rapidlypitching a blade of this length can create torsional damage.

Many actuators are currently being developed for the control ofindividual turbine blades at time constants quicker than turbulent andgust wind events. Some of these active rotor blade technologies include:active blade pitch, microtabs, plasma actuators and passive morphingsurfaces. In some of these control technologies it is unclear what willbe the observed measurement used to determine how to drive the actuatorsby the control system, or to indicate an assessment of damage in amonitoring system. Various embodiments of the present invention includethe use of inertial measurements as a means to estimate both static anddynamic rotor loading.

Some embodiments of the present invention pertain to techniques forestimating the physical static and dynamic loads to a wind turbine rotorblade for use in control algorithms, for use in monitoring systems, anddata to be used in design improvements. The static loads are estimatedfrom the elastic deformation of the blade. Dynamic loads are estimatedthrough modal filtering of the dynamic response.

Some embodiments of the present invention pertain to instrumentationsystems that can achieve some or all of the following aspects:

(1) Select and integrate a sensing system for passively measuring thestatic and dynamic response of the blade;

(2) Identify the effects of variability in the environment and boundaryconditions over the lifecycle of the blade;

(3) Develop and apply algorithms for loads estimation, modal parameterestimation, operating deflection shape determination, and detection ofdamage due to changes in the mechanical properties of the blade.

Currently developing technologies for the development of smart rotorblades include viable sensor networks and control actuators. A smartblade with integrated sensors has been designed, fabricated and groundtested that includes accelerometers, strain gages, temperature sensorsand fiber optic sensors. One aspect of some embodiments of the presentinvention pertains to the design of a sensor array that couldpotentially estimate the loading (L), deflection (δ), and hub forces (F)and moments (M) within the 0-100 Hz range of an operating rotor blade,as shown in FIG. 1 a.

FIG. 1a shows a wind turbine 20 according to one embodiment of thepresent invention. A static support 22 supports above the ground a powerunit 24 that includes an electrical generator for converting mechanicalpower to electrical power. The power unit 24 includes a hub 26 that isrotatable about an axis of rotation 28. A plurality of blades 30 (30 a,30 b, and 30 c) extend outwardly from hub 26 and are attached to hub 26.In one embodiment, blade 30 is a flexible, rotatable member wind turbineblade attached to hub 26. However, in other embodiments of the presentinvention the blade (or rotating member) refers to a helicopter bladeattached to a hub.

Further, other embodiments of the present invention pertain to themonitoring and measurement of any rotating member, especially thoserotating members that are fixed with regards to their attachment to ahub (i.e., the end of the blade cannot translate relative to the hub).In yet other embodiments, there is an assumption about the attachment ofthe member to the hub that the angle of attachment (the slope going intothe hub) does not change as the blade bends. In yet other embodiments,the other end of the rotating flexible member is free to deflect (i.e.,it is not pinned to another structure). In yet other embodiments,another assumption regards the free end of the rotating member beingunable to resist an applied moment.

Referring again to FIG. 1a , a computer 80 is shown in communication(such as electrical or optical communication) with sensors, and in someembodiments with actuators, of wind turbine 20. In one embodiment,computer 80 includes memory that contains an algorithm for monitoringthe health of wind turbine 20. As one example, the algorithm can includethe functions 60 or 160 and derivative 70 described later, either asshown or preferably in reduced form. As will be described herein, areal-time measurement made on a blade 30 can be used in the derivative70 to solve for one or more coefficients of that equation. The algorithmcan further take those coefficients and use them in function 60 or 160to calculate in real-time the loaded geometry of the blade.

In yet other applications, computer 80 can provide a controllingfunction. In such applications, calculation of the geometricconfiguration of the blade (and likewise, estimations of wind loading)can be used in implementation of various control strategies.

In various embodiments of the present invention, an accelerometer 40 isattached at a predetermined position (a station) along the length of theblade. In some embodiments, accelerometer 40 is a DC type accelerometerscapable of simultaneously measuring both constant and dynamicacceleration over the 0-100 Hz bandwidth of an operating blade rotor.Traditional ICP type accelerometers are capable of measuring dynamicacceleration and less capable of measuring static acceleration becausethe piezo-electric measurement element eventually will discharge when aconstant force (acceleration acting on a proof mass) is applied. Evenso, some versions of accelerometer 40 are of the ICP type that have asufficiently long discharge constant. Yet other embodiments of thepresent invention include an accelerometer capable of near-DCmeasurement, especially when the output response is supplemented with alow pass filter or other signal processing.

It is understood that use of the term “DC” does not only refer tocompletely static measurements, but can also include measurements atvery low frequencies (and one embodiment, frequencies less than 10hertz; in a more preferred embodiments, frequencies less than 1 hertz).The DC MEMS measurement element that was selected for this work uses astrain measurement of a deforming diaphragm that is not susceptible todischarge when a constant acceleration is applied. Various embodimentsof the present invention contemplate the use of any type of measurementelement capable of providing a signal corresponding to a constant ornear-constant acceleration, including velocity measurement devices.

DC accelerometers were available in both single axis (PCB 3711) 40′ andtriaxial (PCB 3713) 40 configurations as shown in FIG. 2. Triaxialsensors were predominantly used in the blade so that the response in thespan, lead-lag and flap directions could be measured. Previous workshowed that out-of-plane motions were useful for loads and deflectionestimation and in-plane measurements were useful for damage detection.

The tip of a cantilevered blade is a good sensor location because itobserves all blade modes. However, one aspect of some embodiments of thepresent invention pertains to the installation of sensors within theblade and stations greater than 8 m did not have sufficient interiorclearance to fit a triaxial sensor. Instead, the principal sensorlocation used was the 8 m station at the end of the shear web as shownprior to the final buttoning up of the blade in FIG. 3. At this point,two redundant triaxial sensors were placed in case one failed. A singleaxis sensor measuring in the flap direction was also placed at the samespan-wise distance but shifted toward the trailing edge in the lead-lagdirection. The purpose of this sensor was to estimate the rotationalresponse of the blade. If the blade was rotating then the flapmeasurement of the single axis sensor and triaxial sensor would havebeen out of phase. If there was no rotation then measurements taken withthe two sensors would have been in phase. The sensitivities of thesensors (mv/g) were selected based on the expected magnitude of thecentripetal acceleration, likely the largest acceleration of the blade.

Although various types of sensors and specific ways of mounting thosesensors are shown and described, it is understood that the presentinvention is not so limited. In one example, other embodiments of thepresent invention include the application of sensors (including to butnot limited to accelerometers) on the outer surface of a blade. Yetother embodiments pertain to sensors that are small enough to beembedded within the material of the blade.

In the rotor blade, mode shape deflection magnitude was shifted towardsthe tip where there was decreased stiffness caused by a decreasing crosssectional area in the span direction. By moving the tip group of sensorsinboard, some modes were unobservable because the location was situatedcloser to nodes of vibration. For example, although the 8 m location wassensitive to the first bending mode, it was near a node of the secondbending mode, which in turn could not be observed. To overcome thispotential deficiency a similar group of sensors was placed at the 6.5 mstation as shown in FIG. 2. From previous modal analysis results, the6.5 m station was near resonances of modes that the 8 m station couldnot observe. Therefore, at least one of the two groups of sensorsobserved all of the modes of interest in the 0-100 Hz bandwidth.

The 6.5 m group contained two triaxial sensors and a single axis sensorin an arrangement detailed in FIG. 4. This arrangement was used tomeasure rotation about the span axis and linear motion in all threeprincipal directions. The two triaxial sensors 40 were used to determineif there was a difference of the lead-lag and span in-plane motionsbetween the high-pressure and low-pressure sides due to states oftension and compression, respectively.

Another group of sensors was placed at the 1.74 m station to monitor themotion of the blade at the maximum chord. At this location the generallycircular section of the root transitioned to the outboard airfoilsection. The transition dominated the mode shapes in this region and wasa sensitive measurement of blade operational shape.

A final triaxial sensor was placed in the root close to the hub tomonitor the connection interaction. At this connection the loads fromthe rotor blade were transmitted to the hub and from there onto thedrivetrain and other rotor blades. Additionally, the feedback from theother rotor blades, bearings, gearbox and generator all entered theblade at this location. Measurements at this location accounted for theloads that were not caused by the wind and rotation.

The sensor array installed within this blade can be more comprehensivethan the sensor array installed on blades according to other embodimentsof the present invention. For example, one embodiment of the presentinvention further includes a blade 30 having a single, two axisaccelerometer 40 located at a predetermined location along the length ofblade 40 between the hub and the tip. In addition, it is preferable thatthe measurement device be attached to the blade such that theorientation of the measurement device does not change relative to theblade. Preferably, the two axes of measurement lie generally within theplane of rotation of blades 30. In other embodiments, the axes of thesensors each have a directional component that maps into the plane ofrotation. The purpose of the large sensor array was to prove that thefollowing approaches worked and to then determine the minimum sensorrequirements for each approach. The minimum sensor requirements would beused to design an industrial sensor array.

Methods and apparatus according to one embodiment of the presentinvention are capable of being used with the Sandia NationalLaboratories Micon 65/13 research turbine at the USDA site in Bushland,Tex., shown in FIG. 6a . This turbine was modeled with FAST toinvestigate fitting tip deflection estimators. The tip deflection wasestimated for wind speeds from 1 m/s to 30 m/s and then used to estimatethe best coefficients in equation (17) over the entire tip deflectionrange. The error in tip deflection between the fitted tip estimator andactual tip deflection was used as a metric to compare the different tipestimators.

The Micon 65/13 turbine is a fixed pitch, fixed speed 100 kW turbine.The turbine characteristics are shown in Table 6b. The turbinesimulation was run for a period of 200 seconds with a sample rate of 250Hz. Uniform constant wind loading was applied to the turbine, withsimulated wind speeds spaced every 1 m/s between 1 m/s and 30 m/s. Thehub-height wind speed, turbine power, blade accelerations and tipdeflections were estimated with each simulation. The simulated bladeaccelerations were in the flap, lead-lag and span directions at spandistances of 2 m, 6.5 m and 8 m, which were close to the locations ofthe sensors installed within the smart rotor blade, presented in FIG. 2.

TABLE 6b 9.6144 Distance from rotor apex to blade tip (meters). 0.6144Distance from rotor apex to blade root (meters). 22 Tower height(meters). −4 Rotor shaft tilt (degrees, negative upwards). 4 Coningangle (degrees, positive downwind). 100 Rated power (kW) 55 Constantspeed (rpm)

The following discussion overviews one implementation of rotor bladeestimators for quasi-static deflection and loading estimation, modaldecomposition for dynamic monitoring of wind turbines, passive damagedetection with in plane dynamic displacement measurements, active damagedetection using the method of virtual forces, and temperaturecompensation of frequency response functions of wind turbine rotorblades.

It is beneficial to accurately estimate the static and dynamic loadingof wind turbine rotors. However, there are no practical sensorsavailable for directly measuring load. Consequently, a method was neededto merge the measurement of the sensor with a technical approach thatcould estimate the loading to the rotor. In the following a method isdeveloped assuming that the quasi-static and dynamic acceleration atdiscrete points along the rotor blade can be measured during operation.The following discussion will describe a novel method for themeasurement of blade rotational deformation at each sensor location andthe estimation of blade deflection and loading with the localmeasurement of blade rotation deformation. Additionally, the modalcontributions of a dynamically excited rotor blade are tracked by modalfiltering for fatigue accumulation. However, it is understood that thediscussion herein regarding static and quasi-static measurements andanalysis are separate from some of the aspects of dynamic measurementand analysis discussed herein. Although some embodiments of the presentinvention pertain to monitoring systems that include both static orquasi-static analysis, and also dynamic analysis, yet other embodimentspertain to separate applications of these concepts.

One embodiment of the present invention pertains to the estimation ofmean wind loading and the use of an inertial sensor to measure rotorblade deflection and centrifugal acceleration. Various embodiments ofthe present invention can include any or all of at least three aspects:

(1) The distributed loading of the mean wind and increased velocity winddue to blade rotation cause the blade to elastically curve.

(2) During the curvature of the rotor blade the centripetal accelerationat the reference sensor remains perpendicular to the axis of rotation.

(3) The curvature of the blade causes the surface tangent and normal torotate.

As seen in FIGS. 5B and 5C, the deformation angle 46 (θ) can beestimated from the difference between the centripetal acceleration andthe surface mounted reference sensor. The deformation is proportional tothe distributed mean loading caused by the wind.

The application of accelerometers to wind turbines has been difficult inprevious efforts because the large centripetal acceleration producedfrom the blade rotation has generally interfered with the desireddynamic accelerations. In this work, centripetal acceleration was usedas a strong reference measurement signal for calculating bladedeflection. In one embodiment, a multi axis DC accelerometer placed on ablade can provide an inertial reference vector 44. Vector 44 pointstoward axis of rotation 28, even when blade 30 is deflecting andchanging the orientation of the attached accelerometer.

In FIG. 5a , a cantilevered beam is shown with a fixed base and abi-axial sensor mounted at the tip that measures the inplane andtransverse accelerations. A centripetal acceleration component isincluded from the blade rotating about the x direction at the root, as afunction of radius, R, and rotational speed, ω. In FIG. 5b thequasi-static loading, pstatic(z), from steady wind and the dynamicloading, Pdynamic(z,t), from turbulence, wind shear, etc. causes thebeam to elastically deform. As the beam deforms, the sensor measurementsrotate with the beam. As shown in FIG. 5c , the rotation of the sensorchanges with the rotating beam but the direction of the centripetalacceleration does not change. This is because centripetal accelerationis an inertial effect. Therefore, the local rotation of the sensor, andalso of the beam at that point, can be estimated from the magnitude ofthe ratio of the u and w measurements. From this approach DCaccelerometers that are capable of measuring constant acceleration canbe employed along a wind turbine rotor blade to estimate the localstatic rotational deformation.

Although rotational speeds and radii of the blade can change, thedirection of the centripetal acceleration 44 does not change. Thedirection of the centripetal acceleration is always directed toward theaxis of rotation. This inertial vector 44 can be calculated by a vectorsummation of the measured accelerations w and u. In one embodiment, oneof the measured accelerations is substantially tangent to the surface ofblade 30 (z′ as shown in FIG. 5c ).

When the wind turbine is at rest, the DC component of accelerationmeasured by accelerometer 40 will be oriented to point toward the centerof the gravitational field. However, as the turbine rotates, thecentripetal acceleration vector becomes discernable (from a vectorsummation of other components) and points toward the axis of rotation.As the blade 30 begins to bend, the axis of the attached accelerometer40 likewise begins to change slope. As best seen in FIG. 5c , theincluded angle between the centripetal acceleration vector 44 and thetangential acceleration of the attachment point is also the slope of theblade at the attachment point, and thus a function of the load and thedeflection of the blade. It should be noted that the centripetalacceleration vector 44 can be compared to any measured axis ofacceleration, so long as that direction of measurement is at leastpartly in the plane of rotation of the blade.

As long as the direction of centripetal acceleration remains constantthe sensors always have a reference from which to measure the rotationaldeformation. Furthermore, the estimate of the rotation is from the ratioof the two signals, u and w, and changing amplitude of centripetalacceleration will affect each of those signals equally. The ratio willremain constant regardless of the differences in the centripetalacceleration magnitude.

Although the local rotation (or slope) of the rotor blade can beacquired, the slope itself was not a quantity that could be exactlyrelated to the wind loading. One embodiment of the present inventionpertains to a method developed to produce the deflection and loadingalong the beam as a function of local slope.

FIG. 6 shows a two dimensional representation of the wind turbineloading problem. A wind turbine rotor is connected to a rigid hub 26,denoted by the distance R. The rotor blade itself is flexible anddeformable with, in the case of an Euler-Bernoulli beam, bendingstiffness, El(z), and mass per unit length, m(z), both functions of thespan distance, z, and a length, L. The deflection of the blade in theflap direction, u(z,U), is a function of the span distance and mean windspeed, U. The distributed loading applied to the rotor blade is complexin shape and can be defined as a function of span distance and windspeed. The rotational speed of the rotor blade is denoted Ω.

FIG. 6 also shows some aspects of how blade 30 is modeled and analyzed.In one embodiment, blade 30 is attached to hub 26 such that there is norelative translation between the hub and proximate end of blade 30. Inthis case, hub end condition 62.1 can be stated that the end of blade 30is attached to hub 26. Further, a simplified condition 62.2 that can beapplied to the connection of the hub and the blade is that there is norelative rotation between the hub and the proximate end of the blade,such that the blade is rigidly attached and cannot bend at the junctionof hub and blade root.

Yet other simplifying conditions can be applied to the free end of blade30. One such condition 64.1 is that the free end is free to deflect. Yetanother condition 64.2 is that the end of the blade cannot resist amoment. In some embodiments of the present invention, both of the hubend conditions and both of the free end conditions are applied to themodeling analysis that follows. However, yet other embodiments of thepresent invention do not require all four of these end conditions.

Given the Euler-Bernoulli beam in FIG. 6, Hamilton's variationalprinciple was applied to the potential energy, kinetic energy andloading functions to derive the equation of motion for the beam withconstant mass, constant bending stiffness, and rotational stiffening:

$\begin{matrix}{{{\rho\; A\frac{\partial^{2}u}{\partial t^{2}}} + {{EI}\frac{\partial^{4}u}{\partial z^{4}}} + {\frac{1}{2}\rho\; A\;{\Omega^{2}\left( {{2\left( {R + z} \right)\frac{\partial u}{\partial z}} - {\left( {\left( {R + L} \right)^{2} - \left( {R + z} \right)^{2}} \right)\frac{\partial^{2}u}{\partial z^{2}}}} \right)}}} = {{P_{static}(z)} + {P_{dynamic}\left( {z,t} \right)}}} & (1)\end{matrix}$where all quantities were defined in FIG. 6. For the followingdiscussion only constant loading was assumed so the general staticdeflection equation of the beam was:

$\begin{matrix}{{{{EI}\frac{\partial^{4}u}{\partial z^{4}}} + {\frac{1}{2}\rho\; A\;{\Omega^{2}\left( {{2\left( {R + z} \right)\frac{\partial u}{\partial z}} - {\left( {\left( {R + L} \right)^{2} - \left( {R + z} \right)^{2}} \right)\frac{\partial^{2}u}{\partial z^{2}}}} \right)}}} = {P_{static}(z)}} & (2)\end{matrix}$

Assuming that the wind loading was defined as a summation of linearcombinations of powers of span distance and mean wind speed, thedeflection along the beam can take the form of a power series expansion.Therefore one choice for a deflection estimator basis function 60 cantake the form of a polynomial power series expansion:

$\begin{matrix}{\hat{u} = {{C_{0} + {C_{1}z} + {C_{2}z^{2}} + {C_{3}z^{3}} + {C_{4}z^{4}\mspace{14mu}\ldots} + {C_{n}z^{n}}} = {\sum\limits_{j = 0}^{n}\;{C_{j}z^{j}}}}} & (3)\end{matrix}$where û denotes the deflection estimator, z is the span-wise distance asdefined in FIG. 6, C_(j) are the unknown coefficients, and n is thenumber of terms included in the estimator. Although what has been shownand described is the use of a deflection function 60 in the form of apolynomial series expansion, the present invention is not so limited.Yet other embodiments of the present invention include other types ofseries expansions, including rotating phasor series expansions andothers. Further, the deflection û and x′ both represent deflection (thelatter having been used in FIGS. 5b and 5c ).

At first assumptions were made independent of the distributed loading.For example, regardless of the loading condition it was assumed that therotor blade was fixed at the root and free at the tip, which led to thefollowing boundary conditions, respectively:

$\begin{matrix}{{{{\hat{u}❘_{0}} = 0},{{\frac{\mathbb{d}\hat{u}}{\mathbb{d}z}❘_{0}} = 0},{{{{EI}\frac{\mathbb{d}^{2}\hat{u}}{\mathbb{d}z^{2}}}❘_{L}} = 0}}{{{{and}\mspace{14mu}{EI}\frac{\mathbb{d}^{3}\hat{u}}{\mathbb{d}z^{3}}}❘_{L}} = 0}} & (4)\end{matrix}$

Substituting these boundary conditions into Equation (3) yielded generalsolutions for the first four coefficients independent of the number ofterms included in the estimator:

$\begin{matrix}{{C_{3} = {\sum\limits_{j = 4}^{n}\;{{- \frac{1}{6}}{j\left( {j - 1} \right)}\left( {j - 2} \right)(L)^{j - 3}C_{j}}}}{C_{2} = {\sum\limits_{j = 4}^{n}\;{\frac{1}{2}{j\left( {j - 1} \right)}\left( {j - 3} \right)(L)^{j - 2}C_{j}}}}{C_{1} = 0}{C_{0} = 0}} & (5)\end{matrix}$

This showed that the first four coefficients in the estimator 60 weredependent on the boundary conditions 62.1, 62.2, 64.1, and 64.2. Toutilize the measurement of local rotor blade deformation rotation asdetailed in the previous section, the estimator should have at leastfive terms. However, the present invention also contemplates thoseembodiments in which other types of series are utilized and differentboundary conditions (at the hub and free end) are utilized, such thatthe resultant estimator can have a different number of terms anddifferent types of terms.

In one embodiment, the derivative of estimator 60 is taken with regardsto the position along the blade, which by definition is the slope alongthe blade. To include the local slope estimate from the sensor, acondition was set such that at the sensor location the true rotor bladeslope was the same as the slope of the estimator:

$\begin{matrix}{{{{\frac{\mathbb{d}\hat{u}}{\mathbb{d}z}❘_{z_{1,\ldots}}} = {{\frac{\partial u}{\partial z}❘_{z_{1,\ldots}}} = {{\sum\limits_{j = 4}^{n}\;\left\{ {{{j\left( {j - 1} \right)}\left( {j - 3} \right)L^{j - 2}z} - {\frac{1}{2}{j\left( {j - 1} \right)}\left( {j - 2} \right)L^{j - 3}z^{2}} + {jz}^{j - 1}} \right\}}❘_{z_{1},\ldots}C_{j}}}}{{\frac{\mathbb{d}\hat{u}}{\mathbb{d}z}❘_{z_{1,\ldots}}} = {{\sum\limits_{j = 4}^{n}{{g\left( {L,z,j} \right)}C_{j}}}❘_{z_{1},\ldots}}}}\mspace{236mu}} & (6)\end{matrix}$where u is the actual displacement of the beam and z₁, z₂, . . . are thesensor locations. The two terms on either side of the first, left-mostequality of relationship (6) indicates an assumption that the slope ofblade 30 predicted by estimator 60 is equal to the actual slope of blade60. The right side of the second equality within equation (6) is thederivative 70 of estimator 60.

The first two coefficients of this expression were already defined interms of coefficients greater than n=3. Referring to equation (6), themeasured slope can be substituted onto the left-hand side of equation(6). It is now possible, with this measured slope, to calculate theremaining coefficients C.

Now that the remaining unknowns of estimator 60 have been determined, itis possible to substitute those coefficients into equation (6), alongwith values of z and j, and the deflection of the blade u can becalculated. From this equation an estimator 60 resulted withcoefficients that included the boundary conditions and the restrictionthat the slope condition was satisfied at each sensor location. Thedeflection estimator was now fully defined, excluding where to properlyposition the sensor measurements.

To optimize the sensor placement the error between the deflectionestimator and actual deflection had to be minimized. The error, E,between these two functions over the rotor span (z=0 to L) was definedas:

$\begin{matrix}{E = {{\int_{z = 0}^{L}{\left\lbrack {{u(z)} - {\hat{u}(z)}} \right\rbrack^{2}{w(z)}\ {\mathbb{d}z}}} = {E\left( {U,z_{1},z_{2},\ldots}\; \right)}}} & (7)\end{matrix}$where w(z) is an estimate of static loading. The resulting errorequation is a function of sensor location(s) and wind speed. Minimums inthis function represented optimal locations of sensors for a given windspeed. An error function can be used to design a blade monitoring systemand provide the optimal estimate of blade deflection and loading in themost critical wind regions. Since the blades can be manufactured withthe sensors included, the selection of sensor locations can be performedprior to operation.

The fourth derivative of the deflection estimator, which is a functionof the fourth and higher coefficients, is an estimate of the distributedloading

$\begin{matrix}{{{EI}\frac{\mathbb{d}^{4}\hat{u}}{\mathbb{d}z^{4}}} = {{{EI}{\sum\limits_{j = 4}^{n}\;{{j\left( {j - 1} \right)}\left( {j - 2} \right)\left( {j - 3} \right)z^{j - 4}C_{j}}}} = {\hat{W}(z)}}} & (9)\end{matrix}$

This section has shown how a slope measurement along the blade, althoughnot directly related to the loading and deflection, can be implementedin a manner to estimate the deflection and loading along the rotor bladeby optimizing the sensor location.

The general approach with regards to multiple sensors is to either usemultiple sensors to better estimate C₄ or to expand the estimatorequation to include more degrees of freedom/curvature. For the case ofproducing a better estimate of C₄, the solution would be expanded rowwise:

$\begin{matrix}{C_{4} = {\begin{bmatrix}{{12\; L^{2}z_{1}} - {12\;{Lz}_{1}^{2}} + {4\; z_{1}^{3}}} \\{{12\; L^{2}z_{2}} - {12\;{Lz}_{2}^{2}} + {4\; z_{2}^{3}}} \\{{12\; L^{2}z_{3}} - {12\;{Lz}_{3}^{2}} + {4\; z_{3}^{3}}}\end{bmatrix}^{+}\begin{Bmatrix}{\frac{\partial u}{\partial z}❘_{z_{1}}} \\{\frac{\partial u}{\partial z}❘_{z_{2}}} \\{\frac{\partial u}{\partial z}❘_{z_{3}}}\end{Bmatrix}}} & (11)\end{matrix}$and a least squares/pseudo-inverse (+) approach would be used tocalculate the coefficient.

An example of a two sensor deflection estimator is:

$\begin{matrix}{{\hat{u}(z)} = {\left( {{6\; L^{2}z^{2}} - {4\;{Lz}^{3}} + z^{4}} \right)\frac{1}{4}\frac{\begin{matrix}{{\left( {{3\; L^{2}z_{1}} - {3\;{Lz}_{1}^{2}} + z_{1}^{3}} \right)\frac{\partial u}{\partial z}}❘_{z_{1}} +} \\{{\left( {{3\; L^{2}z_{2}} - {3\;{Lz}_{2}^{2}} + z_{2}^{3}} \right)\frac{\partial u}{\partial z}}❘_{z_{2}}}\end{matrix}}{\begin{matrix}{\left( {{3\; L^{2}z_{1}} - {3\;{Lz}_{1}^{2}} + z_{1}^{3}} \right)^{2} +} \\\left( {{3\; L^{2}z_{2}} - {3\;{Lz}_{2}^{2}} + z_{2}^{3}} \right)^{2}\end{matrix}}}} & (12)\end{matrix}$where the sensor location is used to weight each slope contribution andthe estimator becomes a multivariate optimization problem. This problemwould also include symmetry as the optimal sensor location is validregardless of which optimal location is assigned to each sensor.Additionally the degree of the estimator (n) could be increased by onefor each sensor added and then the matrix in Equation (10) would beexpanded column-wise with a column for each new coefficient of theestimator. The coefficients in Equation (5) were formulated in asummation notation to reduce the difficulty in expanding this matrixcolumn-wise.

The Fatigue, Aerodynamics, Structures, and Turbulence Code (FAST) wasused to estimate realistic loading to a CX-100 rotor blade for anillustrative example of the methods proposed. FAST is a free softwareprogram provided by the National Renewable Energy Laboratory for theanalysis of wind turbines. A model was built, as previously reported, topredict the loading to an operating CX-100 blade attached to a Micon65/13 wind turbine in steady wind loading. The distributed loading inthe flap direction is shown in FIG. 7a , where it is plotted along theblade length from low to high wind speeds. This plot shows that theloads increase linearly along the rotor blade up to around 10 m/s afterwhich the blade enters the stall region. In this region drag andturbulence dominate leading to the highest loads in the widest portionsof the blades between 2 and 5 m.

The distributed loading was analyzed with a statistical softwarepackage, SAS©, to develop a linear regression model consideringindependent and linear combinations of wind speed and span distance upto sixth powers. The linear model, shown in FIG. 7b , that resulted is:P _(flap)(z,U)=b ₁ Uz+b ₂ U ² z ³ +b ₃ U ³ z ⁴ +b ₀  (13)where W is the flap distributed loading, U is the wind speed, z is thespan distance and the coefficients are:

$\begin{matrix}{{b_{1} = {11.823\frac{kg}{m^{2}s}}},{b_{2} = {{- 9.6473} \times 10^{- 3}\frac{kg}{m^{5}}}},{b_{3} = {{2.2305 \times 10^{- 5}\frac{{kg} \cdot s}{m^{7}}\mspace{14mu}{and}\mspace{14mu} b_{0}} = {{- 89.542}\frac{N}{m}}}}} & (14)\end{matrix}$

Table 1 shows the statistical analysis of variance results for thismodel with only three degrees of freedom. The total sum of square is ameasure of the variation in the data and the model sum of squares is theamount of variation that can be accounted for by Eq. (13). The model wasable to account for 94.5% of the total variation of the actualdistributed loading.

TABLE 1 Linear regression model results from SAS, showing 3 DF modelaccounted for 94.5% of the variation. Source DF Sum of Square Model 342770837 Error 596 2500635 Total 599 45271472

This function for distributed loading was used to calculate thedeflection along a beam of constant bending stiffness, mass density andzero rotational speed:

$\begin{matrix}{{u = {\frac{1}{EI}\left\lbrack {{\frac{1}{120}b_{1}{Uz}^{5}} + {\frac{1}{840}b_{2}U^{2}z^{7}} + {\frac{1}{1680}b_{3}U^{3}z^{8}} + {\frac{1}{24}b_{0}z^{4}} + {\frac{1}{6}C_{1}z^{3}} + {\frac{1}{2}C_{2}z^{2}}} \right\rbrack}}\mspace{79mu}{C_{1} = {{{- \frac{1}{2}}b_{1}{UL}^{2}} - {\frac{1}{4}b_{2}U^{2}L^{4}} - {\frac{1}{5}b_{3}U^{3}L^{5}} - {b_{0}L}}}\mspace{79mu}{C_{2} = {{{- \frac{1}{6}}b_{1}{UL}^{3}} - {\frac{1}{20}b_{2}U^{2}L^{5}} - {\frac{1}{30}b_{3}U^{3}L^{6}} - {\frac{1}{2}b_{0}L^{2}} - {C_{1}L}}}} & (15)\end{matrix}$where the complex loading resulted in eighth order span terms. From thetrue deflection equation with nonzero inner radius, the deflectionestimator is preferably designed for the application of a single sensor.FIG. 8a illustrates the error that results from this deflectionestimator as a function of sensor position and wind speed. This plotillustrates more complex features as those previously given, although aminimum error “valley” can be clearly seen near a sensor position of 5 macross all wind speeds.

One method to verify this observation and estimate a numerical value forthe sensor position that minimizes the error, was to integrate the errorfunction across all wind speeds to get error as a function of sensorposition alone, as shown in FIG. 8b . This function suggests that asensor placement at 5 m was the optimal sensor location for thedeflection estimate. It is also important to note that in this case theerror was integrated across all wind speeds, but in some cases it may bemore important to minimize the error in narrow bands of wind speeds orby weighting multiple wind speed bandwidths differently. There is noobstacle to selecting optimal sensor placement with different criteria;however, this method provides the guidance with which to optimize thesensor location regardless of the specific criteria.

An optimized deflection estimator was compared with the actualdeflection as a function of span position and wind speeds. Thedifference between these two functions is shown in FIG. 9a . This figureshows that the deflection estimator over predicts the tip displacementand under predicts the mid-span deflection at higher wind speeds. Highercurvature is expected at higher wind speeds based on the distributedloading function. The maximum deflection error of 0.001 m was only 0.5%percent of the deflection at that point. As discussed in Equation (9)the deflection estimator could also predict the distributed loading asshown in FIG. 9b . The loading distribution compared favorably with FIG.7a , especially when considering that for a single sensor estimator theonly term remaining in the estimator after the fourth derivative was C4.

In another embodiment of the present invention, a deflection estimatorbasis function 160 includes a set of linearly independent equations wasderived that could be easily adapted for any combination of dataavailable and sensor measurements. The general equation of tipdeflection for fitted data with an indefinite number of linear expansionterms of rotation (m) and sensor locations (n) was:

$\begin{matrix}{\delta_{tip} = {{\left\{ {\theta_{1}\mspace{14mu}\theta_{1}^{2}\mspace{14mu}\ldots\mspace{14mu}\theta_{1}^{n_{1}}} \right\}_{1 \times n_{1}}\begin{Bmatrix}C_{1,1} \\C_{2,1} \\\vdots \\C_{n_{1},1}\end{Bmatrix}} + {\left\{ {\theta_{2}\mspace{14mu}\theta_{2}^{2}\mspace{14mu}\ldots\mspace{14mu}\theta_{2}^{n_{2}}} \right\}_{1 \times n_{2}}\begin{Bmatrix}C_{1,2} \\C_{2,2} \\\vdots \\C_{n_{2},2}\end{Bmatrix}\mspace{14mu}\ldots} + {\left\{ {\theta_{m}\mspace{14mu}\theta_{m}^{2}\mspace{14mu}\ldots\mspace{14mu}\theta_{m}^{n_{m}}} \right\}_{1 \times n_{m}}\begin{Bmatrix}C_{1,m} \\C_{2,m} \\\vdots \\C_{n_{m},m}\end{Bmatrix}}}} & (16)\end{matrix}$where δ_(tip) was the tip deflection in units of length, θ_(m) was therotation of the sensor at location m, and C_(n,m) is the coefficient ofθ¹ for rotation at sensor location m. Note that this equation could beadjusted for any number of sensor locations and for different expansionterms for each sensor.

To solve for the coefficients, equation (16) was formed into a matrixfor p data sets that related deflection and sensor rotationmeasurements:

$\begin{matrix}{\begin{Bmatrix}\delta_{1} \\\delta_{2} \\\vdots \\\delta_{p}\end{Bmatrix} = {\quad\left\lbrack {\begin{matrix}\theta_{1,1} & \theta_{1,1}^{2} & \cdots & \theta_{1,1}^{n_{1}} \\\theta_{2,1} & \theta_{2,1}^{2} & \cdots & \theta_{2,1}^{n_{1}} \\\vdots & \vdots & \ddots & \vdots \\\theta_{p,1} & \theta_{p,1}^{2} & \cdots & \theta_{p,1}^{n_{1}}\end{matrix}❘{\begin{matrix}\theta_{1,2} & \theta_{1,2}^{2} & \cdots & \theta_{1,2}^{n_{2}} \\\theta_{2,2} & \theta_{2,2}^{2} & \cdots & \theta_{2,2}^{n_{2}} \\\vdots & \vdots & \ddots & \vdots \\\theta_{p,2} & \theta_{p,2}^{2} & \cdots & \theta_{p,2}^{n_{2}}\end{matrix}❘{\begin{matrix}\cdots \\\cdots \\\cdots \\\cdots\end{matrix}\left. \quad{❘\begin{matrix}\theta_{1,m} & \theta_{1,m}^{2} & \cdots & \theta_{1,m}^{n_{m}} \\\theta_{2,m} & \theta_{2,m}^{2} & \cdots & \theta_{2,m}^{n_{m}} \\\vdots & \vdots & \ddots & \vdots \\\theta_{p,m} & \theta_{p,m}^{2} & \cdots & \theta_{p,m}^{n_{m}}\end{matrix}} \right\rbrack\begin{Bmatrix}C_{1,1} \\\vdots \\\underset{\_}{C_{n_{1},1}} \\\vdots \\\overset{\_}{C_{1,m}} \\\vdots \\C_{n_{m},m}\end{Bmatrix}}}} \right.}} & (17)\end{matrix}$In this formulation all rotation information was included in a singlematrix and all deflection measurements were contained in a vector.Equation (17) was written in a compact form as follows:{δ}_(p×1)=[θ]_(p×N) {C} _(N×1) w/N=(n ₁ +n ₂ + . . . n _(m))  (18)

The pseudo-inverse was used to obtain the least squares estimate for thecoefficients C:{C} _(N×1)=(([θ]_(N×p) ^(T)[θ]_(p×N))_(N×N) ⁻¹[θ]_(N×p)^(T))_(N×p){δ}_(p×1)  (19)

For the case of p>N, equation (16) was an over-determined problembecause there were more measurements of deflection and rotation than thenumber of coefficients to be estimated. Conversely for the case of p<N,more coefficients were estimated than independent deflection androtation measurements available. In application due to constraints oftime and effort, only a certain number of deflection and rotationmeasurements were available that limited the number of coefficients thatwere effectively estimated in equation (17).

While the inventions have been illustrated and described in detail inthe drawings and foregoing description, the same is to be considered asillustrative and not restrictive in character, it being understood thatonly the preferred embodiment has been shown and described and that allchanges and modifications that come within the spirit of the inventionare desired to be protected.

What is claimed is:
 1. A method of predicting the deflection of a windturbine blade operating in a wind, comprising: providing the windturbine blade having two ends and a length therebetween with one endattached to a rotatable hub, an actuator for controlling the windturbine blade, a controller in communication with the actuator, and anaccelerometer attached to the wind turbine blade at a predeterminedstation along the length; expressing the lateral deflection of the windturbine blade as a function of the length of the wind turbine blade;rotating the wind turbine blade by the wind about the hub; measuring theacceleration of the wind turbine blade with the accelerometer duringsaid rotating; using the measured acceleration and establishing aninertial reference with the measured acceleration; using the inertialreference and the function to predict the deflection of the wind turbineblade; and controlling the wind turbine blade with the actuator by thecontroller using the predicted deflection.
 2. The method of claim 1wherein the one end is attached to the hub such that the slope of thewind turbine blade at the hub is about zero during said rotating.
 3. Themethod of claim 1 which further comprises converting the measuredacceleration by at least one of time-based filtering or numericalscaling.
 4. The method of claim 1 which further comprise converting themeasured acceleration to a variable within a derivative of the function.5. The method of claim 4 wherein the accelerometer is a two-axisaccelerometer, and the measured acceleration is converted to an angle.6. The method of claim 1 wherein the function is a series expansionhaving an order greater than
 1. 7. The method of claim 1 wherein thefunction is a series expansion.
 8. The method of claim 7 wherein theseries is a polynomial series.
 9. The method of claim 8 wherein theseries has an order greater than two.
 10. The method of claim 9 whereinthe accelerometer is a first accelerometer and said providing includes asecond accelerometer attached to the wind turbine blade at a differentpredetermined station, said measuring includes responses from the firstaccelerometer and the second accelerometer, which further comprises:measuring the acceleration of the wind turbine blade with the secondaccelerometer during said rotating; wherein establishing an inertialreference is with the measured first acceleration and the measuredsecond acceleration.
 11. The method of claim 7 wherein the series is aseries of rotating phasors.
 12. The method of claim 1 which furthercomprises taking a derivative of the function with respect to length,and wherein said using the measured acceleration includes using thederivative.
 13. The method of claim 1 wherein the length of the windturbine blade extends in a direction substantially orthogonal to therotational axis of the hub.
 14. The method of claim 1 which furthercomprises using the predicted deflection to estimate the wind loading onthe wind turbine blade.
 15. The method of claim 1 wherein the predicteddeflection is used to estimate damage to the wind turbine blade.
 16. Themethod of claim 15 wherein said measuring the acceleration is atfrequencies below about one hertz.
 17. The method of claim 1 wherein theaccelerometer is a two-axis accelerometer, and the measured accelerationis converted to an angle.
 18. The method of claim 1 wherein theaccelerometer is a first accelerometer and said providing includes asecond accelerometer attached to the wind turbine blade at a differentpredetermined station, said measuring includes responses from the firstaccelerometer and the second accelerometer, which further comprises:measuring the acceleration of the wind turbine blade with the secondaccelerometer during said rotating; wherein establishing an inertialreference is with the measured first acceleration and the measuredsecond acceleration.
 19. The method of claim 1 wherein said measuringthe acceleration is at frequencies below about ten hertz.
 20. The methodof claim 1 wherein said measuring the acceleration is at frequenciesbelow about one hertz.
 21. The method of claim 1 wherein measuring theacceleration is tangent to the surface of the wind turbine blade. 22.The method of claim 1 wherein the other end of the wind turbine blade isfree.